Properties of Dynamical Systems on Dendrites and Graphs

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id="M1">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id="M2">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id="M3">\begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>

Compact attractors of an antithetic integral feedback system have a simple structure

Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a two-dimensional linear system (making the closed-loop system a nonlinear four-dimensional system) has been analyzed in much detail. This system has a unique equilibrium but, depending on parameters, it may exhibit periodic orbits. An interesting open question is whether other dynamical behaviors, such as chaotic attractors, might be possible for some parameter choices. This note shows that, for any parameter choices, every bounded trajectory satisfies a Poincaré-Bendixson property. The analysis is based on the recently introduced notion of k-cooperative dynamical systems. It is shown that the model is a strongly 2-cooperative system, implying that the dynamics in the omega-limit set of any precompact solution is conjugate to the dynamics in a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors.

The Distributional Asymptotics Mod 1 of (logb n)

This paper studies the distributional asymptotics of the slowly changing sequence of logarithms (logb n) with b ∈ 𝕅 \ {1}. It is known that (logbn) is not uniformly distributed modulo one, and its omega limit set is composed of a family of translated exponential distributions with constant log b. An improved upper estimate (\sqrt {\log N} /N) is obtained for the rate of convergence with respect to (w. r. t.)the Kantorovich metric on the circle, compared to the general results on rates of convergence for a class of slowly changing sequences in the author’s companion in-progress work. Moreover, a sharp rate of convergence (log N/N)w. r. t. the Kantorovich metric on the interval [0, 1], is derived. As a byproduct, the rate of convergence w.r.t. the discrepancy metric (or the Kolmogorov metric) turns out to be (log N/N) as well, which verifies that an upper bound for this rate derived in [Ohkubo, Y.—Strauch, O.: Distribution of leading digits of numbers, Unif. Distrib. Theory, 11 (2016), no.1, 23–45.] is sharp.

The Omega limit set of a family of chords

In this paper, we study the limit behavior of a family of chords on compact energy hypersurfaces of a family of Hamiltonians. Under the assumption that the energy hypersurfaces are all of contact type, we give results on the Omega limit set of this family of chords. Roughly speaking, such a family must either end in a degeneracy, in which case it joins another family, or can be continued. This gives a Floer theoretic explanation of the behavior of certain families of symmetric periodic orbits in many well-known problems, including the restricted three-body problem.

Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

For a Denjoy homeomorphism $f$ of the circle $S$, we call a pair of distinct points of the $\omega$-limit set $\omega (\,f)$ whose forward and backward orbits converge together a gap, and call an orbit of gaps a hole. In this paper, we generalize the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalized Sturmian system. Moreover, for any Denjoy homeomorphism $f$ having a finite number of holes and for any transitive orientation-preserving homeomorphism $f_1$ of the circle with the same rotation number $\rho (\,f_1)$ as $\rho (\,f)$, we construct a family $f_\varepsilon$ of Denjoy homeomorphisms of rotation number $\rho (\,f)$ containing $f$ such that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f), f)$ for $0<\varepsilon <\tilde{\varepsilon }<1$, but the number of holes changes at $\varepsilon =\tilde{\varepsilon }$, that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f_{\tilde{\varepsilon }}), f_{\tilde{\varepsilon }})$ for $\tilde{\varepsilon }\leqslant \varepsilon <1$ but $\lim _{\varepsilon \nearrow 1}f_\varepsilon (t)=f_1(t)$ for any $t\in S$, and that $f_\varepsilon$ has a singular limit when $\varepsilon \searrow 0$. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.

On the finiteness of attractors for piecewise maps of the interval

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.